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Navigates conceptual possibility spaces by synthesizing glass bead games, world hopping, and ordered locales via observational bridges with Narya-verified transitions.
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> *"The bead connects. The bridge directs. The hop observes."*
Synthesizes concepts from mathematics, music, and philosophy via category theory morphisms, Badiou triangle, and world-hopping mechanics. Useful for interdisciplinary exploration.
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"The bead connects. The bridge directs. The hop observes."
Glass Hopping synthesizes three skills into one:
| Skill | Contribution | Type-Theoretic Role |
|---|---|---|
| glass-bead-game | Conceptual connections (beads) | Objects in frame |
| world-hopping | Possibility navigation (hops) | Morphisms between worlds |
| ordered-locale | Directional structure (≪) | Bridge types |
The key insight: World hops are bridge types in an ordered locale.
Bead₁ ────Bridge(B₁, B₂)────→ Bead₂
│ │
↓ observational ↓
World₁ ←────U ≪ V────→ World₂
A hop from world W₁ to W₂ is an observational bridge type:
def Hop (W₁ W₂ : World) : Type := Bridge W₁ W₂
The bridge is:
Each bead corresponds to an open in the ordered locale:
class GlassBead:
domain: str # mathematics, music, philosophy
concept: str # The conceptual content
open_set: FrozenSet # Points in the locale where bead is "active"
trit: int # GF(3) polarity: -1, 0, +1
The frame of opens forms a complete Heyting algebra:
The Badiou triangle inequality is automatically satisfied by the ≪ order:
If U ≪ V and V ≪ W, then U ≪ W (transitivity)
Distance becomes bridge composition length:
def bridge_distance(W1, W2, locale):
"""Distance = minimum bridge chain length"""
# Find shortest path in ≪ graph
opens = [U for U in locale.frame.carrier if W1.active_in(U)]
for U in opens:
for V in locale.frame.carrier:
if locale.order_ll(U, V) and W2.active_in(V):
return 1 # Direct bridge exists
return float('inf') # No bridge path
game = GlassHoppingGame(
locale=triadic_gf3(), # Ordered locale for structure
seed=0x42D, # Deterministic randomness
players=3 # Triadic (GF(3) conserved)
)
move = GlassHop.Place(
bead=Bead(domain="mathematics", concept="prime"),
open_set=frozenset([1]), # Active in the "plus" region
trit=+1
)
# Creates: Open in frame with bead attached
move = GlassHop.Bridge(
from_bead=bead_prime,
to_bead=bead_harmony,
bridge_type="harmonic_series" # Primes → overtones
)
# Creates: WayBelow relation in ordered locale
# Verifies: Open cone condition
move = GlassHop.Hop(
from_world=current_world,
via_bridge=bridge_prime_harmony,
to_world=target_world
)
# Executes: Badiou event along bridge
# Preserves: GF(3) conservation
# Validates: Triangle inequality
move = GlassHop.Observe(
world=superposed_world,
observable=bead_measurement
)
# Collapses: Multiple possible states to one
# Bridge type: Observational equality witness
| Move | Points | Bridge Bonus | GF(3) Bonus |
|---|---|---|---|
| PLACE | 10 | — | ×2 if balances |
| BRIDGE | 25 | ×2 if ≪ verified | ×3 if cone-preserving |
| HOP | 50 | ×(1/distance) | ×2 if conserved |
| OBSERVE | 30 | ×2 if unique | — |
Elegance: Shorter bridge chains score higher.
def World : Type := sig (
seed : Nat,
epoch : Nat,
state : State,
invariants : List Invariant
)
def Bead (W : World) : Type := sig (
domain : Domain,
concept : String,
active : W .state → Prop
)
def GlassHop (W₁ W₂ : World) (B₁ : Bead W₁) (B₂ : Bead W₂) : Type := sig (
bridge : Bridge W₁ W₂,
bead_transfer : (x : W₁ .state) → B₁ .active x → B₂ .active (transport bridge x),
trit_conserved : trit B₁ + trit B₂ + trit bridge ≡ 0 (mod 3)
)
def OpenConeCondition (L : OrderedLocale) (U : Open L) : Type := sig (
up_is_open : IsOpen L (up_closure L U),
down_is_open : IsOpen L (down_closure L U)
)
def TriangleInequality (W₁ W₂ W₃ : World)
(h₁₂ : GlassHop W₁ W₂) (h₂₃ : GlassHop W₂ W₃) : Type :=
sig (
composed : GlassHop W₁ W₃,
distance_bound : distance composed ≤ distance h₁₂ + distance h₂₃
)
Each glass hop preserves the triadic invariant:
Σ trits = trit(bead₁) + trit(bridge) + trit(bead₂) ≡ 0 (mod 3)
| Trit | Role | Voice | Glass Hop Action |
|---|---|---|---|
| -1 | VALIDATOR | Anna (German) | Verify bridge types |
| 0 | COORDINATOR | Amélie (French) | Navigate locale |
| +1 | GENERATOR | Luca (Italian) | Create new beads |
╔═══════════════════════════════════════════════════════════════╗
║ GLASS HOPPING: Observational Bridge Navigation ║
║ Seed: 0x42D | Players: 3 | GF(3): Conserved ║
╚═══════════════════════════════════════════════════════════════╝
Turn 1 [PLUS/Luca]: PLACE(bead="prime numbers", open={1}, trit=+1)
→ Bead placed in PLUS region
→ Points: 10
Turn 2 [ERGODIC/Amélie]: BRIDGE(prime → harmony, type="overtone series")
→ Bridge created: {1} ≪ {0,1}
→ Open cone verified ✓
→ Points: 25 × 2 = 50
Turn 3 [MINUS/Anna]: HOP(world_math → world_music, via=bridge_overtone)
→ Event: "Harmonic Analysis"
→ Triangle inequality: d(math,music) ≤ d(math,physics) + d(physics,music) ✓
→ GF(3): (-1) + (0) + (+1) = 0 ✓
→ Points: 50 × 2 = 100
Turn 4 [PLUS/Luca]: OBSERVE(world_music, bead="Ramanujan's taxicab")
→ Collapsed: 1729 = 1³+12³ = 9³+10³
→ Observational bridge: number ↔ harmony
→ Points: 30
Total: 190 points
GF(3) Sum: 0 ✓
Bridge Chain: prime ≪ overtone ≪ taxicab
from ordered_locale import OrderedLocale, Frame, triadic_gf3
# Create glass hopping locale
locale = triadic_gf3()
# Beads are opens
bead_validator = frozenset([-1, 0, 1]) # Full locale
bead_coordinator = frozenset([0, 1]) # Ergodic + Plus
bead_generator = frozenset([1]) # Plus only
# ≪ order is hop direction
assert locale.order_ll(bead_validator, bead_coordinator)
assert locale.order_ll(bead_coordinator, bead_generator)
from sheaves import DirectionalSheaf
# Game state as sheaf over locale
game_sheaf = DirectionalSheaf(locale=locale)
# Sections carry bead data
game_sheaf.add_section(
bead_validator,
{"role": "VALIDATOR", "beads": [...]}
)
# Restrictions respect ≪
game_sheaf.add_restriction(
bead_validator,
bead_coordinator,
lambda state: {k: v for k, v in state.items() if k != "role"}
)
from ordered_locale import points_functor, spatialization
# Extract worlds from locale
worlds = points_functor(locale)
# Each world = completely prime filter
for w in worlds:
print(f"World {w}: filter = {w.filter_elements}")
# Spatialization recovers point-based topology
spatial = spatialization(locale)
# Start glass hopping game
just glass-hop
# Single move
just glass-hop-move place "prime" +1
# Bridge creation
just glass-hop-bridge "math/prime" "music/harmony" "overtone"
# Hop execution
just glass-hop-hop world_1 world_2 bridge_name
# Verify triangle inequality
just glass-hop-triangle w1 w2 w3
# Full demo
just glass-hop-demo
~/.agents/skills/glass-hopping/
├── SKILL.md # This file
├── glass_hopping.py # Python implementation
├── glass_hopping.ny # Narya bridge types
├── game_state.py # Sheaf-based state
└── demo.py # Interactive demo
glass-bead-game — Conceptual synthesisworld-hopping — Badiou event navigationordered-locale — ≪ order and frame theorynarya — Bridge type verificationtriad-interleave — GF(3) schedulingunworld — Derivational chainsSkill Name: glass-hopping Type: Synthesis / Navigation / Verification Trit: 0 (ERGODIC - mediator between beads and hops) GF(3): Conserved at each hop Bridge Types: Observational (asymmetric, directed) Triangle Inequality: Enforced by ≪ transitivity
This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:
general: 734 citations in bib.duckdbThis skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:
Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826
The skill participates in triads satisfying:
(-1) + (0) + (+1) ≡ 0 (mod 3)
This ensures compositional coherence in the Cat# equipment structure.